Large values of quadratic Dirichlet $L$-functions

Abstract

Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $\hat{I}\text{c◯}$-results for the family of quadratic Dirichlet $L$-functions $L(\ddot{I}ƒ,\ddot{I}{‡}_d)$, where $d$ runs over all fundamental discriminants with $|d|\le X$ and $\ddot{I}ƒ\in [1/2,1]$ is fixed. This study advances understanding of the maximum size of $L(\ddot{I}ƒ,\ddot{I}{‡}_d)$ within the segment $\ddot{I}ƒ\in [1/2,1]$. In particular, we improve upon Soundararajan's results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed $\ddot{I}ƒ\in (1/2,1]$.

26 pages; comments are welcome